Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R Statement I: Change in internal energy of a system containing n mole of ideal gas can be written as ΔU = n CV (Tf - Ti) = nR/(γ - 1) (Tf - Ti), where γ = Cp/Cv, Ti = initial temperature, Tf = final temperature. Statement II: Relation between degree of freedom f and γ (= Cp/CV) is γ = 1 + 2/f Choose the correct answer from the options given below

The correct answer is Option 2. Full mathematical solution steps and hints are accessible in the interactive StemCET workspace.

Physics Q32 – JEE-MAIN 2026

← Back

Home Physics ThermodynamicsQuestion #510

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R Statement I: Change in internal energy of a system containing

n

mole of ideal gas can be written as

Δ U = n C_{V} (T_{f} - T_{i}) = \frac{n R}{γ - 1} (T_{f} - T_{i})

, where

γ = \frac{C_{p}}{C_{v}}

T_{i}

= initial temperature,

T_{f}

= final temperature. Statement II: Relation between degree of freedom

f

and

γ (= C_{p} / C_{V})

γ = 1 + \frac{2}{f}

Choose the correct answer from the options given below

Clue 1 of 3: Internal Energy of Ideal Gas

Recall the fundamental formula for the change in internal energy of an ideal gas and the definitions of specific heats.

🧠 Full Solution Path

Step 1: Evaluate Statement I (Assertion A)✦ Active

Statement I gives the change in internal energy of an ideal gas as $Δ U = n C_{V} (T_{f} - T_{i}) = \frac{n R}{γ - 1} (T_{f} - T_{i})$ . The first part, $Δ U = n C_{V} (T_{f} - T_{i})$ , is the fundamental definition for an ideal gas. To verify the second part, we use Mayer's relation $C_{p} - C_{V} = R$ and the definition $γ = \frac{C_{p}}{C_{V}}$ . From these, we get $C_{p} = γ C_{V}$ . Substituting into Mayer's relation: $γ C_{V} - C_{V} = R \Rightarrow C_{V} (γ - 1) = R \Rightarrow C_{V} = \frac{R}{γ - 1}$ . Substituting this into the internal energy formula yields $Δ U = n (\frac{R}{γ - 1}) (T_{f} - T_{i})$ . Thus, Statement I is True.

Step 2: Evaluate Statement II (Reason R)○ Expand

Statement II provides the relation between the degree of freedom $f$ and $γ$ as $γ = 1 + \frac{2}{f}$ . For an ideal gas, the molar specific heat at constant volume is $C_{V} = \frac{f}{2} R$ . Using Mayer's relation, $C_{p} = C_{V} + R = \frac{f}{2} R + R = R (\frac{f}{2} + 1) = R (\frac{f + 2}{2})$ . The adiabatic index $γ = \frac{C_{p}}{C_{V}} = \frac{R (\frac{f + 2}{2})}{R (\frac{f}{2})} = \frac{f + 2}{f} = 1 + \frac{2}{f}$ . Thus, Statement II is True.

Step 3: Determine if R is the correct explanation for A○ Expand

Both Statement I (Assertion A) and Statement II (Reason R) are true. Statement I's second form for $Δ U$ is derived from the definition of $γ$ and Mayer's relation ( $C_{V} = \frac{R}{γ - 1}$ ). Statement II gives a fundamental relation between $γ$ and the degrees of freedom $f$ . While related to ideal gas properties, Statement II does not directly explain the derivation of the internal energy formula in terms of $γ$ as presented in Statement I. Therefore, R is not the correct explanation for A.

💡 Teacher's Secret Hint

Distinguish between a true statement and a correct explanation. A correct explanation should logically justify the assertion.

✦ STEM Console utilizes AI models to generate step-by-step explanations and math clues. AI can make mistakes.

Physics Question 32 – JEE-MAIN 2026

Clue 1 of 3: Internal Energy of Ideal Gas

🧠 Full Solution Path